# How does air pressure affect the speed of sound?

How does air pressure affect the speed of sound in relation to its kinetic theory etc?

I have tried searching this but i have not found a suitable answer as other websites simply related to air pressure waves.

• Ideally, it doesn't. It does however affect the loudest possible sound. Speed of sound is a function of temperature. Nov 11, 2014 at 22:16
• Nov 12, 2014 at 9:43

To first order, the speed of sound is not affected by pressure. Pressure waves can be shown to fulfill the D'Alembert wave equation $$(c_S^2\,\nabla^2 - \partial_t^2)\psi=0$$ where the wavespeed $$c_S$$ is given by:

$$c_S = \sqrt{\frac{K}{\rho}}$$

where $$K$$ is the bulk modulus of the medium in question and $$\rho$$ its density. Now, for an ideal gas, the bulk modulus $$K$$ is in most conditions proportional to the pressure; if the compression is adiabatic (good approximation for high frequency sound, as there is little time for heat to shuttle back and forth in the gas), then $$K=\gamma\,P$$, where $$\gamma$$ is the Heat Capacity Ratio or Adiabatic Index. However, from the ideal gas law $$P\,V=n\,R\,T$$ we have:

$$\rho = \frac{n\,M}{V} = \frac{P\,M}{R\,T}$$

where $$M$$ is the mean molar mass of the gas in question in kilograms. Thus the pressures cancel out in the speed of sound:

$$c_S = \sqrt{\frac{\gamma\,R\,T}{M}}$$

Thus we see that the speed is also weakly affected by the humidity - more water in the air lowers the mean molecular mass. If we put $$M=0.029$$, $$T=300K$$ and $$\gamma = 1.4$$ for air, we get $$c_S=347{\rm m\,s^{-1}}$$.

• Wilhelm van Schaik is correct, humidity increases speed of sound. Nov 3, 2021 at 9:46
• @Samson Which is as I’ve also said, as this fact can easily be inferred from my last equation for $c_S$ Nov 3, 2021 at 15:55